Cloth physics : drop/unroll a cloth sheet from the top of a building
Here is another way to go.  Instead of using pinning to connect the bottom vertices of the banner to the bottom cylinder, parent the bottom cylinder to the bottom 3 vertices of the banner.  (Keep the pinning to the top cylinder so the top of the banner stays put.)  That way, as the bottom of the cloth naturally falls due to gravity, the cylinder will fall with it.  You would need to adjust the mesh of the banner to a staring position where the bottom vertices are near the top ones, and probably "bunch up" the rest of the flag so it's all up at or near the top.  

Here is a quick and dirty implementation of that idea: